Homework014
64. Assume that the lifetime of a certain component has an exponential distribution with mean
3.
Find all of the following:
the value of the parameter,
f
, the 55th percentile and the
standard deviation of lifetimes, the probability that a component lasts longer than 7.5 years,
the probability, for a component that has functioned for 1 year, that it will function for at
least three more years.
SOLUTION:
Let X indicate the lifetime of a certain component, i.e. X
∼
Exp(
λ
=
1
μ
=
1
3
).
a. Parameter, i.e.
λ
=
1
3
.
b.
f
, the pdf,
f
(
x
) =
1
3
e
−
x
3
, x
≥
0.
c.
x
.
55
=
ln(1
−
p
)
−
λ
=
ln(1
−
0
.
55)
−
1
3
= 2
.
396.
d. P(X
>
7
.
5) =
e
−
1
3
(7
.
5)
=
e
−
2
.
5
(= 0
.
082).
e. P(X
≥
1 + 3

X
>
1) = P(X
≥
3) =
e
−
1
3
(3)
=
e
−
1
(= 0
.
368).
65. Do Exercise 6 on p.270.
SOLUTION:
Let X indicate the waiting time of a certain component, i.e. X
∼
Exp(
λ
= 1).
a. P(X
>
5) =
e
−
(1)(5)
=
e
−
5
(= 0
.
007).
b. Based on a., 5 minutes is an unusual long time to wait (0.007 is too small).
c. If one really waited for 5 minutes until the next hit, such unusual event happens most
likely because the claim is wrong (i.e mean is supposed to be higher).
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 Fall '11
 DANIELCONUS
 Statistics, Normal Distribution, Probability, Standard Deviation, Variance, Probability theory, bin, certain component

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